 # Advanced Math Help with Any Problems! excefebraxp 2022-09-07

## Discrete math one to one questionUse the pigeonhole principle to prove that if $n\ge 2$ people go to the same party, there are at least 2 people who shake hands with the same number of other people.Hint: Take the set of people at the party as your domain, define a function that evaluates how many people each person shook hands with. peckishnz 2022-09-07

## Proof by cases - discrete mathI need to prove by the cases technique that: moidu13x8 2022-09-07

## Find the tight upper and lower bounds for the following recurrence: Felix Cohen 2022-09-07

## Discrete Math problem combinations and restrictionsI am trying to get my head around an idea but I can't seem to get it to work.Imagine you have the word "MAMMAL"Lets see I wanted to figure out how many ways I could rearrange the letters. Well that is easy. It is simply$6!/3!2!=60$ possibilitiesBut what if I added a restriction to it such that all M's must be together.Then I could consider all M's as one letter and it would be $4!/2!=12$ possibilities.Again I understand that. But what if the restriction was there needs to be a minimum of 2 M's together at all times.If I were to do the same process as the previous example, I would combine 2 M's into one. Which then would become $5!/2!=60$.This seems to be a wrong answer because it is the same as my first calculation of finding all possibilities without any restrictions. Can anyone please explain to me as to how I need to approach the last problem of finding number of combinations where at least 2 M's are always together? Mutignaniz2 2022-09-07

## Discrete Math Construct Tree from WeightsConsider the weights: 10, 12, 13, 16, 17, 17.(a) Construct an optimal coding scheme for the weights given by using a tree.(b) What is the total weight of the tree you found in part (a)? Staffangz 2022-09-07

## Upperbound on harmonic number discrete mathHow do I show ${H}_{{2}^{k}}\le k+1$ for each $k\ge 0$? Dulce Cantrell 2022-09-07

## I've been given the recursive function:$T\left(n<=2\right)=O\left(1\right)$With $n={2}^{{3}^{p}}$ where p is a positive integer. tamola7f 2022-09-07

## Discrete MathI am trying to understand a question I got:Let U be a set.For every $A\subseteq U$ we define the function: ${f}_{A}:P\left(U\right)\to P\left(U\right)$ such as fA(B)=(A∩B).Prove: f is injective and surjective if and only if $A=U$First, I do not understand what ${f}_{A}$ and ${f}_{A}\left(B\right)$ mean.I think that if I will understand this, I will be able to start solving the question.Any tips will be wonderful! metal1fc 2022-09-07

## Arranging books in bookshelves with the capacity of each shelf givenThere are k identical bookshelves in which each shelf cannot contain m or more books. In how many ways can n distinct books be arranged on these k bookshelves?If there is no condition on the capacity of each shelf, the number of ways to arrange books equals $\sum _{i\in \left[k\right]}L\left(n,i\right)$, where L(n,i) denotes the Lah number. However, because of that constraint, I have trouble in solving the problem.I tried several ways to solve this problem by separating the cases via (1) the number of shelves which contains the full-number of books, or (2) the number of non-empty shelves. For the second trial, I observed that, if j denotes the number of non-empty shelves, then the number of ways to arrange the books is zero if $j<⌊n/m⌋$.However, these methods does not proceed quite well, since it looks like these methods result in the recurrence relation rather than the exact form of the number.For the related concepts, I have studied Catalan number, (both signed and unsigned) first and second Stirling number, Bell and Lah number, and the integer partition.Any insight or comment are welcomed. Gauge Odom 2022-09-07

## How to prove that subset at odd size is equal to subset at even size?I'm trying to prove that:$\sum _{i=0}^{⌈\frac{n}{2}⌉}\left(\genfrac{}{}{0}{}{n}{2i}\right)=\sum _{i=0}^{⌈\frac{n}{2}⌉}\left(\genfrac{}{}{0}{}{n}{2i+1}\right)$if n is odd, I can do it with the identity: $\left(\genfrac{}{}{0}{}{n}{k}\right)=\left(\genfrac{}{}{0}{}{n}{n-k}\right)$, because if we have odd number like 7 we have 4 ($=⌈\frac{n}{2}⌉$) pairs: (7,0),(6,1),(5,2),(4,3). The left element is odd and the right one is even. So:$\left(\genfrac{}{}{0}{}{7}{0}\right)=\left(\genfrac{}{}{0}{}{7}{7}\right),\phantom{\rule{thinmathspace}{0ex}}\left(\genfrac{}{}{0}{}{7}{6}\right)=\left(\genfrac{}{}{0}{}{7}{1}\right),...$But when I'm trying to do this about even numbers (like 8), I can't use this method.So my question is: What can I do at cases of even numbers? Jimena Hatfield 2022-09-07

## A tight bound for $T\left(n\right)={2}^{n}T\left(\frac{n}{2}\right)+{n}^{n}$Given recurrence$T\left(n\right)={2}^{n}T\left(\frac{n}{2}\right)+{n}^{n}$How we can show that $T\left(n\right)\le {n}^{n}$?I show that $T\left(n\right)\ge {n}^{n}$ because of existence the term ${n}^{n}$ in T(n). Alfredeim 2022-09-07

## a is a factor of 10 and b is a factor of 15.$\begin{array}{|cc|}\hline \text{Column A}& \text{Column B}\\ \text{The smallest value that ab must be a factor of}& 30\\ \hline\end{array}$Is Column A greater than/ smaller than / equal to Column B? Jazmyn Saunders 2022-09-07

## Help with summations for discrete mathI could start them but I have no idea how to simplify after expanding them.$\sum _{i=28}^{n}\left(3{i}^{2}-4i+\left(5/{7}^{i}\right)\right)$$\sum _{i=1}^{n}\frac{i}{{2}^{n-i+1}}$ mashingcho9v 2022-09-07

## I have an upper and lower bound number:upper: 21lower: 3I then have a second number that can be anywhere between this range, I would like the second number to increment faster when it is closer to the lower bound and slower when it is closer to the upper bound.How can I achieve this using mathematics? Zackary Duffy 2022-09-07

## How can i prove this?$\mathrm{\forall }n\mathrm{\exists }p\left({p}^{2}\le n<\left(p+1{\right)}^{2}\right)$The domain of quantifiers is N. ezelsbankuk 2022-09-07

## What is a strongly discrete sequence?What is the definition (and possibly examples or applications) of strongly discrete sequence? I have seen this term, but I am not able to find anything about it. Ciolan3u 2022-09-07

## Discrete math De Morgan's Laws. So like if I have a form ${p}^{q}$ and say I make p equal giving you cookies and q giving you milk so the sentence is just "Giving you cookies and giving you milk" like when I think about it if I think of the opposite(negation) I just automatically think it's not giving you cookies and not giving you milk but this is wrong but is it really? I mean am I just misunderstanding what opposite/negation means here? Kailey Vargas 2022-09-07

## Using Direct Proofs in Discrete MathUse a direct proof to show that if x is a rational number, then ${x}^{2}$ is also a rational number.I know I need to use the definition of rational numbers but don't know how to do it in this problem. atarentspe 2022-09-07
## Validity of arguments in discrete mathAll girls are tall.Anyone who is tall and dark will pass.Claire is a girl.Conclusion: Claire will pass.converting the statements to predicate logic:- $G\left(x\right)=x$ is a girl- $A\left(x\right)=x$ is tall- $B\left(X\right)=x$ is dark- $C\left(x\right)=x$ passes$\phantom{\rule{1em}{0ex}}\forall \left(x\right)\left[G\left(x\right)\to A\left(x\right)\right]\phantom{\rule{0ex}{0ex}}\forall \left(x\right)\left[\left[A\left(x\right)\wedge B\left(x\right)\right]\to C\left(x\right)\right]\phantom{\rule{0ex}{0ex}}G\left(c\right)\to A\left(c\right)$ .....................c stands for claireAm i right till now? and please help me proceed to finish the problem . Ciolan3u 2022-09-07